4i

4i is a purely imaginary complex number, equal to zero plus four times the imaginary unit i, where i is defined by i^2 = -1. In standard rectangular form it is written as 0 + 4i. In the complex plane it lies on the positive imaginary axis at a distance of 4 from the origin.

The modulus and argument of 4i are straightforward: the modulus |4i| is 4, and the argument (angle) is π/2, placing it at the point (0, 4). In polar form it can be expressed as 4e^{iπ/2} or 4cis(π/2). This representation emphasizes its geometric interpretation as a vector of length 4 rotated 90 degrees from the positive real axis.

Algebraically, 4i behaves like a real scalar multiple of i. For example, r(4i) = (4r)i for any real r, and (4i)(4i) = -16. The reciprocal is 1/(4i) = -i/4, and its complex conjugate is -4i. Operations with 4i follow the same rules as complex numbers, with i^2 = -1 governing all multiplications involving i.

Context and usage: 4i frequently appears in solving linear equations, polynomial equations, and in applied fields such as electrical engineering and physics when representing imaginary components of signals and phasors. It is an example of a purely imaginary number, illustrating how the imaginary unit scales along the imaginary axis.